Questions tagged [modular-curves]

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3 votes
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Generalisation of Sharifi's conjecture for Siegel varieties

I recently learned Sharifi's conjecture in this article by F.Takako and K.Kato. According to my understanding, this conjecture states that (the conjugation invariant) of the projective limit $\...
0 votes
0 answers
197 views

Geometric meaning of cusps/component labels in Katz-Mazur book

In Katz-Mazur book "Arithmetic moduli of elliptic curves" there is a very short section (see image below) regarding the cusp-labels and component-labels. The set of cusps labels intuitively ...
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9 votes
1 answer
456 views

Universal elliptic curve and the Tate curve

I've seen the following sentence come up a few times in papers: Let $E$ be the universal elliptic curve over the modular curve $Y_1(N)$. Then the localization of $E$ at any choice of cusp is ...
6 votes
0 answers
188 views

Semistable model of product of modular curves

Does the product $Y_1(Np) \times Y_1(Np)$ admit a semistable model over $\mathbf{Z}_p[\zeta_p]$ with a natural moduli-space interpretation? Less telegraphically: let $p$ be a prime, and $N \ge 4$ ...
7 votes
1 answer
341 views

Questions on the $j$-invariant

The j-invariant as a modular function is typically defined $$j(\tau) = \frac{E_4(\tau)^3}{\Delta(\tau)}$$ since $E_4$ is a modular form of weight 4 and $\Delta$ has weight 12, it follows that $j$ is a ...
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3 votes
1 answer
282 views

Explicit natural correspondence between cusps of $X(N)$ and isomorphism classes of level $N$ structures on Tate($q^N$)

In Katz' paper Antwerp III, section 1.4 (Ka-14) one reads (we assume $n \geq 3$ integer): ''The scheme $\overline{M}_n - M_n$" over $\mathbb{Z}[1/n]$ is finite and étale, and over $\mathbb{Z}[1/n,...
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5 votes
1 answer
187 views

Geometry of the section $X_0(N) \to X_0(pN)$ given by the canonical subgroup

Goren and Kassaei's paper "The Canonical Subgroup: a Subgroup-Free Approach" takes the position that the canonical subgroup of order $p$ for elliptic curves over $\mathbb{Z}_p$ with $\Gamma$-level ...
2 votes
0 answers
215 views

Standard application of Oort-Tate classification theorem

$\DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\tors}{tors}$In Mazur's paper “Modular curves and the Eisenstein ideals”, on the bottom of page 159, it says that if $T$ is a open subscheme of $...
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2 votes
0 answers
119 views

Weighted projective lines and elliptic curves

The modular curves of low level can sometimes be describes as weighted projective lines. For example, over $\mathbb{Z}[1/2]$ the compactified stack of elliptic curves with full level 2 structure is ...
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3 votes
1 answer
229 views

Automorphisms of the modular curve defined over $\mathbb{Q}$

Let $p\geq 3$ be a prime number. The modular curve $X(p)$ can be considered as a connected smooth projective curve over complex numbers. There is a subgroup $\mathrm{PSL}_2(\mathbb{F}_p)$ inside its ...
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1 vote
1 answer
205 views

When is $X_0(N)$ representable?

Fix a base ring $R$. Is the rigidification of the modular "curve" $X_0(N)$ in the sense of Abramovich-Olsson-Vistoli representable iff $N\geq 5$ and $N$ is $0, 2, 3, 6, 8, 11 (\mathrm{mod}\: 12)$? ...
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0 votes
0 answers
77 views

Isomorphism between 2nd symmetric product and Jacobian

Let $X=X_0(N)$ be hyperelliptic with $g(X)\geq 2$ with $\infty$ as a cusp and $\iota$ as the hyperelliptic involution. Then the map $$X^{(2)} \longrightarrow Jac(X)$$ $$D \longrightarrow [D-\infty -\...
2 votes
0 answers
179 views

What is the modular curve for level 1, 2?

An elliptic curve over a scheme $S$ is the data of a proper smooth morphism of schemes $\pi: E\rightarrow S$ whose geometric fibres are connected curves of genus $1$ and a section $e: S \rightarrow E$....
5 votes
1 answer
410 views

When is the conductor of an elliptic modular curve equal to its level?

Suppose the usual modular curve $E=X_0(N)$ over $\mathbb{Q}$ has genus 1 (e.g. $N=15$). Define the conductor of $E/\mathbb{Q}$ as the ideal/integer: $$M=\prod_{p}p^{f(E/\mathbb{Q}_p)},$$ where $$f(...
2 votes
0 answers
152 views

Local-global compatibility and modular curves

I have been told by some people that local-global Langlands compatibility for $GL_2$ (the vanilla version, not the one being developed in this decade by Emerton and others) implies Shimura conjecture ...
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6 votes
0 answers
374 views

Integral models of perfectoid modular curves

Scholze constructed perfectoid modular curve and its canonical and anticanonical part in his paper On torsion in the cohomology of locally symmetric varieties (Annals of Mathematics 182 (2015) pp 945–...
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2 votes
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178 views

Igusa curve at infinite level

In the paper "Le halo spectral" (http://perso.ens-lyon.fr/vincent.pilloni/halofinal.pdf) and in the following paper "The adic cuspidal, Hilbert eigenvarieties" (http://perso.ens-lyon.fr/vincent....
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7 votes
3 answers
488 views

Endomorphism ring of $J_0(p)$ and Hecke operators

Let $J_0(p)$ be Jacobian of the modular curve $X_0(p)$ over $\mathbb Q$ where $p$ is a prime, consider the subring $\mathbb T$ inside $\newcommand{\End}{\operatorname{End}}\End_{\mathbb Q}(J_0(p))$ ...
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10 votes
0 answers
265 views

Generalized Jacobians and modular units

Let $X$ be a proper algebraic smooth curve over a characteristic zero field $k$ and let $J$ be the Jacobian variety of $X$. Let $K$ be the function field of $X$. Assume that we are given $n$ distinct ...